Special Control Charts II Review X-bar, R, S

X-Bar & Sigma-Charts Used when sample size is greater than 10 X-Bar Control Limits: Approximate 3 limits are found from S & table Sigma-Chart Control Limits: Approximate, asymmetric 3 limits from S & table

X-Bar & Sigma-Charts Limits can also be generated from historical data: X-Bar Control Limits: Approximate 3 limits are found from known 0 & table Sigma-Chart Control Limits: Approximate, asymmetric 3 limits from 0 & table

X-bar and S charts Allows us to estimate the process standard deviation directly instead of indirectly through the use of the range R S chart limits: UCL = B6σ = B4*S-bar Center Line = c4σ = S-bar LCL = B5σ = B3*S-bar X-bar chart limits UCL = X-doublebar +A3S-bar Center line = X-doublebar LCL = X-doublebar -A3S-bar

Special Variables Control Charts x-chart for individuals Moving ranged control chart QSUM chart

Individual Measurements IE 355: Quality and Applied Statistics I Individual Measurements Sometimes repeated measures in a subsample don't make sense: inventory level accounts payable – price of an item Other reasons for using individual measurements Variation in sample only reflects measurement error, e.g., batch production of chemicals Automated inspection – every unit is analyzed Production rate very slow – inconvenient to wait for large enough sample (c) 2002 D.H. Jensen

Control Charts for Individual Measurements IE 355: Quality and Applied Statistics I Control Charts for Individual Measurements Notes about Individuals Charts: Sample points must be relatively frequent There is more sampling error (false alarm & insensitivity) Sample points tend to be non-normal points are not averages and central limit theorem does not apply Control Chart Types for Individuals: Shewhart x-chart and Moving Range chart MA – Moving Average chart EWMA – Exponentially Weighted Moving Average CUSUM – Cumulative Sum (c) 2002 D.H. Jensen

Moving Range Control Chart Viscosity of Paint Primer IE 355: Quality and Applied Statistics I Moving Range Control Chart Viscosity of Paint Primer MR2i = |xi – xi-1| or MR3i = |xi – xi-2| Computation of the Moving Range: Obs i xi MR2 MR3 1 33.75 - 2 33.05 0.70 3 34.00 0.95 0.25 4 33.81 0.19 0.76 5 33.46 0.35 0.46 … n Average 33.52 0.48 0.42 (c) 2002 D.H. Jensen

Moving Range Control Chart, Cont'd IE 355: Quality and Applied Statistics I Moving Range Control Chart, Cont'd General model for moving range chart: Plot MR2i = |xi – xi-1| or MR3i = |xi – xi-2| on control chart Substituting estimates for mR and sR and using “3-sigma” limits: Where MR is: (c) 2002 D.H. Jensen

Moving Range Control Chart, Cont'd IE 355: Quality and Applied Statistics I Moving Range Control Chart, Cont'd For MR2 use d2 for n = 2 For MR3 use d2 for n = 3 Very similar to Range chart except we’re using moving range instead of average range (c) 2002 D.H. Jensen

x Control Chart (Individual Measurements Chart) IE 355: Quality and Applied Statistics I x Control Chart (Individual Measurements Chart) Plot sample statistic: x General model for x chart Substituting estimates for mx and sx and using 3-sigma limits where: (c) 2002 D.H. Jensen

IE 355: Quality and Applied Statistics I x Control Chart Cont'd x chart upper and lower limits: (c) 2002 D.H. Jensen

Cautions for x & Moving Range charts: IE 355: Quality and Applied Statistics I Cautions for x & Moving Range charts: Always check x’s for normality If x’s not normal, control limits are inappropriate Use zone rules ONLY if the x’s are Normal Very BAD at detecting small shifts, i.e., shifts < 2s x chart (n = 5) x chart (n = 1) size of shift b ARL1 1s 0.78 5.25 0.98 43.96 2s 0.07 1.08 0.84 6.30 3s 0.00 1.00 0.50 2.00 (c) 2002 D.H. Jensen

IE 355: Quality and Applied Statistics I Montgomery(5th ed.) Example 5-5, p. 250 Viscosity of Aircraft Primer Paint Plot MR2i = |xi – xi-1| on control chart From initial data compute x and MR: (since d2 = 1.128 for n =2) Control Limits for Moving Range chart (use D4 & D3 for n =2) (c) 2002 D.H. Jensen

Ex.: Viscosity of Aircraft Primer Paint, Cont'd IE 355: Quality and Applied Statistics I Ex.: Viscosity of Aircraft Primer Paint, Cont'd Compute control Limits for x Chart (c) 2002 D.H. Jensen

IE 355: Quality and Applied Statistics I CUSUM Control Chart Incorporates all the information in the sequence of sample values by plotting the cumulative sums of the deviations of the sample values from a target value, m0 CUSUM can be used to monitor process mean defectives defects variance CUSUM can have sample size n  1 We concentrate on sample size n = 1 (c) 2002 D.H. Jensen

Basic Principle of CUSUM IE 355: Quality and Applied Statistics I Basic Principle of CUSUM Plot Ci – CUSUM sample statistic Example: Say target m0 = 10 If the process remains in-control, Ci remains near 0 Obs i xi (xi – m0) 1 9.45 -0.55 2 7.99 -2.01 -0.55 – 2.01 = -2.56 3 9.29 -0.71 -2.56 – 0.71 = -3.27 4 11.66 1.66 -3.27 + 1.66 = -1.61 5 12.16 2.16 -1.61 + 2.16 = 0.55 (c) 2002 D.H. Jensen

Tabular CUSUM Control Chart IE 355: Quality and Applied Statistics I Tabular CUSUM Control Chart xi ~ N(m0, s) - quality characteristic CUSUM works by compiling the statistics: Ci+ = accumulated deviations above m0 (resets to 0 if it would go negative) Ci– = accumulated deviations below m0 (resets to 0 if it would go positive) The Tabular CUSUM Record following values in table: where starting values are (c) 2002 D.H. Jensen

Tabular CUSUM Control Chart Cont'd IE 355: Quality and Applied Statistics I Tabular CUSUM Control Chart Cont'd Let m1 = out-of-control value then K is reference value chosen halfway between target m0 and out-of-control value With shift expressed in std dev units, i.e., and accumulate deviations from μ0 that are greater than K and are reset to zero upon becoming negative (c) 2002 D.H. Jensen

How to Determine if Process Out-of-Control? IE 355: Quality and Applied Statistics I How to Determine if Process Out-of-Control? H - decision interval If or exceed the decision interval (H), the process is considered out-of-control Rule of thumb value for H Choose H to be five times the process standard deviation, H = 5s Counters N+ and N– record the number of consecutive periods the CUSUM and rose above zero, respectively. The counters can be used to indicate when the shift most likely occurred (c) 2002 D.H. Jensen

IE 355: Quality and Applied Statistics I Example m0 =10, n =1, s = 1.0 Say magnitude of shift we want to detect is ds = 1 (1.0) = 1.0 Parameters: m1 = m0 + ds = 10 + (1)(1) = 11 K = d/2 = ½ = 0.5 H = 5s = (5)(1) = 5 Counters N+ and N- N+ records the number of consecutive periods since the upper-side cusum Ci+ rose above the value of zero N- records the number of consecutive periods since the lower-side cusum Ci- rose above the value of zero (c) 2002 D.H. Jensen

IE 355: Quality and Applied Statistics I The upper-side cusum at period 29 is C29+ = 5.28 Since this is the first period at which Ci+ > H =5, we would conclude that the process is out of control at that point Since N+ = 7 at period 29, we would conclude that the process was last in control at period 29-7=22, so the shift likely occurred between periods 22 and 23 (c) 2002 D.H. Jensen

Tabular CUSUM Example

Estimate of New Shifted Process Mean IE 355: Quality and Applied Statistics I Estimate of New Shifted Process Mean Use this estimate to bring process back to the target value m0 e.g.: At period 29, New process average estimate is New process average estimate is: Quality characteristic has shifted from 10 to 11.25. Make process adjustment to decrease value of quality characteristic by 1.25. (c) 2002 D.H. Jensen

Notes about CUSUM control charts IE 355: Quality and Applied Statistics I Notes about CUSUM control charts Do not apply zone rules Do not apply run rules Successive values of and are not independent (c) 2002 D.H. Jensen

Recommendations for CUSUM design IE 355: Quality and Applied Statistics I Recommendations for CUSUM design Let H = hs and K = ks where s is process std dev Using h = 4 or h =5 and k = 1/2 gives CUSUM w/ good ARL Shift (multiple of s) ARL for h = 4 ARL for h = 5 168 465 0.25 74.2 139 0.50 26.6 38.0 0.75 13.3 17.0 1.00 8.38 10.4 1.50 4.75 5.75 2.00 3.34 4.01 2.50 2.62 3.11 3.00 2.19 2.57 4.00 1.71 2.01 (c) 2002 D.H. Jensen

CUSUM Chart X-bar, R charts work well to detect significant shifts in the mean (1.5s - 2.0s) However, suppose we wish to detect smaller shifts in the process Consider a piston ring process with target value at 10.0 cm. First 10 sampled at normal with m = 10 Second 10 sampled at normal with m = 11 Control limits computed at LCL = 7, UCL = 13

Example

Example

CUSUM Idea å å Show the cumulative effects of relatively small changes = ( x - m ) i j o j = 1 i - 1 å = ( x - m ) + ( x - m ) i o j o j = 1 = ( x - m ) + C i o i - 1

Example, CUSUM Table

Example; CUSUM Chart